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PROFESSOR: Welcome
back to recitation.

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In this video I'd like us to
work on the following problem.

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We're going to let capital F
of x equal the integral from 0

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to x of little f of t dt.

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And I want us to find the
quadratic approximation

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for big F near x equals 0
in terms of the little f.

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And then I want us to
answer this question.

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What assumptions do we have
to make about little f?

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You'll see we're making some
assumptions about little f.

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I want to know what
those assumptions are.

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So I'm going to give
you a little bit of time

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to work on this problem.

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And then I'll be back and
I will work on it with you.

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Welcome back.

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Hopefully you were able
to make some good headway

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with this problem.

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So again, what we're trying to
do, is we've defined capital F.

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And we want to find the
quadratic approximation for it

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near x equals 0 in
terms of little f.

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Knowing that capital F of x is
equal to the integral from 0

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to x of little f of t dt.

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And then at the end,
we'll talk about what

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are the assumptions we've made.

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So to start off, what
I actually want to do

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is remind us, what is the
quadratic approximation

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for capital F, just
in terms of itself?

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Of capital F?

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So let me come over
to the right and we'll

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write down what the
quadratic approximation is,

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near x equals 0.

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So, this gives us that F
of x is approximately F

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of 0 plus F prime of 0 times
x plus F double prime at 0

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over 2 times x squared.

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This is just the
quadratic approximation

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of capital F at x equals 0.

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That's what it is.

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So now all we need to do,
I've asked you to write it

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in terms of little f.

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So what we need to do is figure
out what each of these things

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actually is.

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What capital F of 0 is, what
capital F prime of 0 is,

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and what capital F
double prime of 0 is,

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in terms of the
function little f.

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So let's see.

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Well first, let's figure
out what capital F of 0 is.

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Let's write down what it is.

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The integral from 0
to 0 of f of t dt.

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Well we know, when you
take the integral from a

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to a of a function,
you should get 0.

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So capital F at 0 is just 0.

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So we have one piece.

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Now, let's figure out
what capital F prime is.

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Well, capital F
prime of x, we know

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using fundamental
theorem of calculus--

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let's just write it down
so we can see it easily--

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we know this is
just little f of x.

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So if I wanted capital F prime
at 0, I just come over here.

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That's just going
to be little f at 0.

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So now, if we look back up
at what we have at the top,

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we filled in this part--
or we know that part,

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that's 0-- we know this part.

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So now all we need is capital
F double prime evaluated

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at x equals 0.

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And then we'll be done.

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Now, how are we
going to do this?

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Well we know what
capital F prime is.

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So let's come in, sort of
up here, and look again.

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We know what capital F prime is.

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We know it's little f.

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So if I want to take a second
derivative, of capital F--

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let me write that down here--
that's going to be taking

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a derivative of little f.

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Right?

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That'll be, that's
d/dx of little f of x.

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So I can just write this
as little f prime of x.

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So again, let me show
you where that came from.

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We knew big F prime
was equal to little f.

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So when I take another
derivative of this,

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I have to take a
derivative of this.

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So that's why capital
F double prime

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is equal to little f prime.

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So we're almost there.

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I think I'm about to run out
of room down here, so let

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me come over a
little bit further.

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So now, we just need
to evaluate this at 0.

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Well, that's just going to be
evaluating little f prime at 0.

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So now I have all the pieces.

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Let's come back in and
fill it in one more time.

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So what do I get?

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I get-- let me, now I
have all the pieces.

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Now to answer the question.

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Little f-- or sorry-- big F of
x, its quadratic approximation

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is going to be big F of
0, which we said was 0.

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So let's come over and look.

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Here's big F of 0.

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I know that's 0.

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And then I'm going to add to
that big F prime at 0 times x.

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That's little f of 0 times x.

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Let's come back
over and do that.

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So we add to that
little f at 0 times x.

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And let's come back.

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One more thing
we're going to need.

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We need one more term.

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That's big F double prime
at 0 over 2 times x squared.

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That I've left big
F double prime at 0

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is little f prime at 0.

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So that's f prime at 0
over 2 times x squared.

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So we had all the pieces.

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We just had to put them in.

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This does not look like an
approximately by the way.

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Sorry.

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There we go.

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So what do we get?

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This expression above
is equal to-- well why

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did I write the 0 again?

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Sorry I was trying to
avoid writing the 0.

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It's equal to f of 0 times x
plus f prime of 0 over 2 times

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x squared.

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So the quadratic approximation
of capital F in terms

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of little f is exactly this.

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Now what assumptions are
we making about little f?

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Well to apply the fundamental
theorem of calculus,

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you need little f is continuous.

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But we actually
need more than that.

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Because what have we done?

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We've actually taken a
derivative of little f.

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So we need that this function,
little f, is actually

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differentiable.

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So that is, that actually
includes, of course,

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that little f is continuous.

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So we don't even have to say
continuous in the assumptions.

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We just have to
say, we're assuming

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little f is differentiable,
at least near 0.

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So let's go back and just
remind ourselves what we did.

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I'm going to go all the
way over, remind ourselves

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of the problem and kind
of bring us back through.

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So we started off with
this definition of big F.

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And I wanted us to find a
quadratic approximation for it

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near x equals 0.

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And then we just mentioned,
what are the assumptions?

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So I reminded you,
first, of what

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is the quadratic approximation
of big F in terms of itself,

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in terms of big F. And
we just write it out.

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And then we had defined what
is this in terms of little f,

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what is this in
terms of little f,

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and what is this in
terms of little f?

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And so what do we look at?

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Well, we evaluate f of 0.

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That's easy enough.

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We found big F prime
in terms of little f.

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That's fundamental
theorem of calculus.

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And then we evaluate
that at x equals 0.

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And then we know that
if we want to take

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a second derivative
of this function,

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we need to take a
derivative of the little f.

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And so we get big F double prime
is actually little f prime.

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So we evaluate that at 0 to
get big F double prime at 0.

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And then we come in and we just
fill in everything we needed.

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And so ultimately, this
is the final answer

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of the quadratic approximation
of big F in terms of little

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f near x equals 0.

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So I think that's
where we'll stop.

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